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Trigonometric Ratios Of Compounds — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios Of Compounds4 Marks
Question
If $\text{a}\cos2\text{x}+\text{b}\sin2\text{x}=\text{c}$ has $\alpha$ and $\beta$ as its roots, then prove that, $\tan\alpha+\tan\beta=\frac{2\text{b}}{\text{a+c}}$
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Answer

$\cos2\theta=\frac{1-\tan^2\theta}{1+\tan^2\theta}$ $\sin^2\theta=\frac{2\tan\theta}{1+\tan^2\theta}$ substitute these valuse in the given equation, it reduces to $\text{a}(1-\tan^2)+\text{b}(2\tan\theta)=\text{c}(1+\tan^2\theta)$ $(\text{c+a})\tan^2\theta+2\text{b}\tan\theta+\text{C-a}=0$ As $\alpha$ and $\beta$ are roots sum of roots, $\tan\alpha+\tan\beta=\frac{2\text{b}}{\text{c+a}}$

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